# How does ECDSA work in Bitcoin

1. Closure: If a and b are in a group G, then a + b is in the group G
2. Associativity: (a + b) + c = a + (b + c)
3. Identity Element: a + 0 = 0 + a = a
4. For every a there exists b such that a + b = 0
5. For Abelian Groups only, commutativity: a + b = b + a
1. The identity element is the point at infinity, 0
2. The inverse of the point P is the one symmetric about the x axis
3. Addition is defined as: given three aligned, non-zero points, P, Q, and R you have P + Q + R = 0. The order does not matter for these three points, so P + (Q+R) = 0, (P+Q) + R = 0, (P+R) + Q = 0, etc. This allows us to prove elliptic curves are both commutative, and associative.
1. Prime modulo of the finite field = 97
2. The elliptic curve described above where a = 2 and b = 3
3. A random point P = (3,6) — this is called the base point
4. The order of the subgroup (set of cyclic points based on P) = 5
1. Prime modulo: 2²⁵⁶ - 2³² - 2⁹ - 2⁸ - 2⁷ - 2⁶ - 2⁴ - 1 → this is a really really big number approximately equal to all of the atoms in the universe. It is also represented in hexadecimal as: FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F
2. Elliptic curve where a = 0 and b = 7
3. Base point P in hexadecimal = 04 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 9C47D08F FB10D4B8
4. Order in hexadecimal = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141

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